SOLUTION: find all the roots of each equation. x^4+4x^3+6x^2+8x+8=0

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Question 7044: find all the roots of each equation.
x^4+4x^3+6x^2+8x+8=0

Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
you need to factorise this: x%5E4%2B4x%5E3%2B6x%5E2%2B8x%2B8=0. Seeing as how they have asked you, and you know next to nothing about solving x%5E4 polynomials, there has to be a "nice" solution for you. So, lets just re-order the polynomial slightly to:

x%5E4%2B6x%5E2%2B8+%2B4x%5E3%2B8x=0

Lets treat the first 3 terms as a quadratic and the last 2 terms we will just factorise normally: So

%28x%5E2%2B2%29%28x%5E2%2B4%29+%2B+4x%28x%5E2%2B2%29+=+0. Both of these terms have a common term, namely %28x%5E2%2B2%29, so we can factorise again to give:

%28x%5E2%2B2%29%28x%5E2+%2B+4+%2B+4x%29+=+0 and the second bracket can be factorised to give

%28x%5E2%2B2%29%28x%2B2%29%28x%2B2%29+=+0

so either x%5E2%2B2+=+0 or x+2 = 0
so either x%5E2+=+-2 or x = -2
so either x+=+%2B+sqrt%28-2%29 or x+=+-+sqrt%28-2%29 or x = -2
--> x+=+%2B+sqrt%282%29i or x+=+-+sqrt%282%29i or x = -2

jon.